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arXiv:2611.01679

Bounds and sufficient conditions in quantum Zeno effect

Vladimir Parfenyev, Sergey Belan

quant-phcond-mat.stat-mechSubmitted Nov 1, 2026• Announced Nov 3, 2026
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Abstract

When a quantum system undergoes a series of repeated measurements during its time evolution to check whether it remains in the initial state, the resulting state lifetime can decrease (the quantum Zeno effect) or increase (the quantum inverse Zeno effect) as the inter-measurement period increases. Several transitions between these two regimes may occur within the same system. A carefully chosen measurement period $τ_\ast$ (or rate $r_\ast$ in the case of a stochastic measurement protocol) brings the state lifetime to its global minimum. Here, we describe the upper and lower bounds on the optimized state lifetime, which are expressed in terms of characteristic time scales encoded in the state survival probability $P(T)$, which is assumed to decrease monotonically to zero at large $T$. In addition, we analyze the bounds on the optimal measurement periods/rates and formulate sufficient conditions for observing the inverse Zeno effect. The analysis is also extended to states with non-monotonic survival functions. The results are illustrated numerically with an example of a system of optically trapped ultra-cold atoms.

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Abstract The effective lifetime of a quantum state can increase (the quantum Zeno effect) or decrease (the quantum anti-Zeno effect) in the response to increasing frequency of the repeated measurements and the multiple transitions between these two regimes are potentially possible within the same system. An interesting question arising in this regards is how to choose the optimal schedule of repeated measurements to achieve the maximal possible decay rate of a given quantum state. Addressing the issue of optimality in the quantum Zeno dynamics, we derive a range of rigorous results, which are, due to generality of the theoretical framework adopted here, applicable to the majority of models appeared in the quantum Zeno literature. In particular, we prove the universal dominance of the regular stroboscopic sampling in the sense that it always provides the shortest expected decay time among all possible measurement procedures. However, the implementation of the stroboscopic protocol requires the knowledge of the optimal sampling period which may depend on the fine details of the quantum problem. We demonstrate that this difficulty can be overcome with the tricky non-regular measurement schedule inspired by the scale-free restart strategy used to speed up the completion of the probabilistic algorithms and Internet tasks in computer science as it allows to achieve a near-optimal decay rate in the absence of detailed knowledge of the underlying quantum statistics. Besides, our general approach reveals unexpected universality displayed by the quantum systems subject to the optimally tuned rate of Poissonian measurements and the simple statistical criteria to discriminate between Zeno and anti-Zeno regimes following from this universality. We illustrate our findings with an example of Zeno dynamics in the system of optically-trapped ultra-cold atoms and discuss the implications arising from them.

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If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, t∝Nα. The limit survival probability exhibits a sudden jump from 1 to 0 at α=1/2, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for α≥1, the limit probability becomes sensitive to the spectral properties of the initial state and to the arithmetic properties of the measurement periods.

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The temporal evolution of an unstable quantum mechanical system undergoing repeated measurements is investigated. In general, by changing the time interval between successive measurements, the decay can be accelerated (inverse quantum Zeno effect) or slowed down (quantum Zeno effect), depending on the features of the interaction Hamiltonian. A geometric criterion is proposed for a transition to occur between these two regimes.

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When a quantum system undergoes a series of repeated measurements during its time evolution to check whether it remains in the initial state, the resulting state lifetime can decrease (the quantum Zeno effect) or increase (the quantum inverse Zeno effect) as the inter-measurement period increases. Several transitions between these two regimes may occur within the same system. A carefully chosen measurement period τ∗τ_\astτ∗​ (or rate r∗r_\astr∗​ in the case of a stochastic measurement protocol) brings the state lifetime to its global minimum. Here, we describe the upper and lower bounds on the optimized state lifetime, which are expressed in terms of characteristic time scales encoded in the state survival probability P(T)P(T)P(T), which is assumed to decrease monotonically to zero at large TTT. In addition, we analyze the bounds on the optimal measurement periods/rates and formulate sufficient conditions for observing the inverse Zeno effect. The analysis is also extended to states with non-monotonic survival functions. The results are illustrated numerically with an example of a system of optically trapped ultra-cold atoms.
Abstract The effective lifetime of a quantum state can increase (the quantum Zeno effect) or decrease (the quantum anti-Zeno effect) in the response to increasing frequency of the repeated measurements and the multiple transitions between these two regimes are potentially possible within the same system. An interesting question arising in this regards is how to choose the optimal schedule of repeated measurements to achieve the maximal possible decay rate of a given quantum state. Addressing the issue of optimality in the quantum Zeno dynamics, we derive a range of rigorous results, which are, due to generality of the theoretical framework adopted here, applicable to the majority of models appeared in the quantum Zeno literature. In particular, we prove the universal dominance of the regular stroboscopic sampling in the sense that it always provides the shortest expected decay time among all possible measurement procedures. However, the implementation of the stroboscopic protocol requires the knowledge of the optimal sampling period which may depend on the fine details of the quantum problem. We demonstrate that this difficulty can be overcome with the tricky non-regular measurement schedule inspired by the scale-free restart strategy used to speed up the completion of the probabilistic algorithms and Internet tasks in computer science as it allows to achieve a near-optimal decay rate in the absence of detailed knowledge of the underlying quantum statistics. Besides, our general approach reveals unexpected universality displayed by the quantum systems subject to the optimally tuned rate of Poissonian measurements and the simple statistical criteria to discriminate between Zeno and anti-Zeno regimes following from this universality. We illustrate our findings with an example of Zeno dynamics in the system of optically-trapped ultra-cold atoms and discuss the implications arising from them.
If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, t∝Nα. The limit survival probability exhibits a sudden jump from 1 to 0 at α=1/2, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for α≥1, the limit probability becomes sensitive to the spectral properties of the initial state and to the arithmetic properties of the measurement periods.
In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such non-selective measurements. Importantly, we show that there is a difference between using non-selective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems, and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective measurements can differ very significantly as compared to the repeated selective measurement scenario.
Quantum Zeno effect can be applied to quantum information processing，and can reveal the nature of quantum measurement. In addition, it has also many potential applications. This suggests that studying the quantum Zeno effect has great theoretical and experimental significance. In this work, the system of a two-level atom interacting with a single mode field is considered and the dynamics of the system subjected to successive projection measurements is studied, and the quantum Zeno effect is presented. Moreover, the influence of the quantum Zeno effect on atomic population inversion is discussed. Based on Schrödinger equation, the survival probability of the initial state of the two-level atom subjected to frequently repeated measurements can be obtained. The survival probability depends on the time interval between measurements. It is seen that the exponential decay of the system under slowly frequent measurements is presented instead of the naturally oscillatory process. For slowly repeated measurements the atomic population inversion and the survival probability of initial state decline rapidly at the early time and then both of them become unchanged. As the time intervals of the measurements are sufficiently short, the quantum Zeno effect occurs. These results have also shown how the measurement can inhibit the atomic population inversion.
The quantum Zeno effect consists in the hindrance of the evolution of a quantum system that is very frequently monitored and found to be in its initial state at every single measurement. On the basis of the correct formula for the survival probability, i.e. the probability of finding the system in its initial state at every single measurement, we critically analyze a recent proposal and experimental test, that make use of an oscillating system. PACS: 03.65.Bz, 42.50.-p 1 The seminal formulation of the quantum Zeno effect, due to Misra and Sudarshan [1], deals with the probability of observing an unstable system in its initial state throughout a time interval ∆ = [0, t]. The purpose of this note is to point out that the quantum Zeno effect has not been experimentally observed, yet, in its original formulation. Indeed, we shall argue that the interesting proposal by Cook [2], that makes use of a two-level system undergoing Rabi oscillations, as well as the beautiful experiment performed by Itano et al. [3], investigate the probability of finding the